For $0\leq k\leq n$,
$$
\mathbb P(X=k\mid X+Y=n)
=\binom nk\left(\frac{\lambda}{\lambda+\mu}\right)^k
\left(\frac{\mu}{\lambda+\mu}\right)^{n-k}.
$$
Thus
$$
\boxed{X\mid(X+Y=n)\sim\operatorname{Binomial}\left(n,\frac{\lambda}{\lambda+\mu}\right).}
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ia / Paper 2 / 3F / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ia/paper-2/3f/ii/solution